Papers

The Straight-Through Estimator (STE) is a widely used heuristic for Quantization-Aware Training (QAT), but its surrogate gradients can exhibit substantial mismatch with the underlying quantized objective, leading to noisy updates and parameter oscillations, particularly in ultra-low-bit regimes. We propose the Quantization-Aware Minimal-Norm Optimizer (Q-MINO), a temporal bundle method that combines gradient consensus, state-drift regularization, and an alignment constraint to construct stabilized, minimum-norm update directions from recent optimization states. Q-MINO solves the resulting constrained subproblem using a warm-started Frank–Wolfe procedure with a feasible fallback initialization. Theoretically, via a stochastic Lyapunov Kurdyka–Łojasiewicz (KL) framework, we show that Q-MINO achieves asymptotic neighborhood convergence. Moreover, we detail numerical experiments with Q-MINO at various quantizations.

For a graph G, the Boolean quadric polytope P(G) is the convex hull of the binary solutions of $y_{ij}=x_ix_j$ for $ij\in E(G)$, and Q(G) is its standard linear relaxation. For series-parallel graphs, Q(G) together with the odd-cycle inequalities describes P(G). Lee and Skipper showed vol(Q(G)) is polynomial-time computable for bounded treewidth and gave a closed formula for vol(P(G)) when G is a cycle. We resolve their question of giving an efficient algorithm for vol(P(G)) on series-parallel graphs. With $d=|V(G)|+|E(G)|$, we compute it with $O(d^7)$ arithmetic operations, and $O(d^5)$ if G is a cactus. The algorithm is a dynamic program over the series-parallel decomposition. The same framework computes vol(Q(G)) (hence the number of linear extensions of the incidence poset of G) with $O(d^4)$ operations, and $O(d^3)$ for cacti. With every $x_v$ fixed at 1/2, the recursion reduces to convolutions of univariate polynomials and computes the cut polytope volume of every series-parallel graph with m edges in $O(m^3)$ operations. We also study how much of Q(G) the polytope P(G) occupies. Short odd cycles matter more than long ones: in every graph, the odd-cycle inequalities of a cycle of length $\ell$ cut off at most a fraction $2^{\ell-1}/\ell!$ of Q(G). Hence vol(P(G))/vol(Q(G)) $\ge 1-\sum_C 2^{|C|-1}/|C|!$ for series-parallel G, with C ranging over its cycles. The ratio does not factor over cycles sharing a vertex: in a flower of $k$ copies of $C_\ell$ it decays like $\rho_\ell^k$ for an explicit rational $\rho_\ell$ below the ratio of $C_\ell$. Nevertheless, the small fractions cut off by many long cycles compound: in the worst case, triangle inequalities, or odd-cycle inequalities up to any fixed length, close no fixed fraction of the gap between Q(G) and P(G), and the volume ratio can be exponentially small in d.

We consider multi-dimensional quasiperiodic Schrödinger operators with monotone sawtooth-type potentials in the perturbative localization regime. We prove that a given gap is closed iff a corresponding (via a natural gap labeling) eigenfunction vanishes at the origin. Using this, we show that for a generic choice of the sampling function $f$, the spectrum of such an operator is a Cantor set, with all possible gaps being open. We also provide a class of functions $f$ for which some gaps are closed. Finally, we observe that certain gaps are more difficult to close than the others.

The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.

We prove that the positive first Dirichlet eigenfunction of every smooth domain with convex boundary in $\mathbb{CP}^n$ invariant under the standard $U(1)\times U(n-1)$-action is strictly log-concave for $n=2,3,4$.

Consideration of frequentist type-I error or power based on certain fixed conditions may not fully embrace the aims of a Bayesian analysis; this is particularly true in settings where informative priors are involved. Calderazzo et al. consider adaptations of a testing procedure involving Bayesian borrowing from external control data. Particular caution may be necessary here in order not to condition on parameters and data simultaneously. We consider their example case and argue that (a) the conditioning scheme may have limited practical implications, and (b) if the main concern are assumption violations, these might be better addressed by adapting the likelihood function.

Let $F$ be a projective variety, let $H$ be a normally generated line bundle on $F$, and let $X\in |H^{\otimes d}|$ be a general projective variety. We establish a uniform criterion, formulated in terms of the graded Betti numbers of the section ring $R(F,H)$, that provides sufficient conditions for every automorphism of $X$ to extend to an automorphism of $F$. This extends the classical results of Matsumura-Monsky on automorphisms of hypersurfaces in projective spaces to a broader class of polarized varieties $(F,H)$, including rational homogeneous varieties of Picard number one, smooth Fano threefolds with their anticanonical polarization, and examples of hyperkähler and Calabi-Yau varieties.

Let $L=-\Delta_k+V$ be a Schrödinger operator associated with the Dunkl Laplacian and a nonnegative potential. We characterize the Hardy space $H_L^1$, defined by the maximal operator of the semigroup generated by $-L$, in terms of the Riesz transforms associated with $L$. Under a local integral condition on the potential and a decay condition on the semigroup, formulated relative to an admissible covering by cubes, we prove that \[ \|f\|_{H_L^1} \asymp \|f\|_{L^1(dw)} +\sum_{j=1}^N \|\widetilde{R}_j f\|_{L^1(dw)}. \] The proof combines an atomic characterization of $H_L^1$ with local Hardy space theory and weighted integral estimates for Dunkl derivatives of the heat kernels. The result applies to arbitrary root systems and nonnegative multiplicity functions.

We develop multifidelity methods for constructing triangular transport maps from samples, when high-fidelity data are scarce but lower-fidelity data are more abundant. Using this set of multifidelity data, we approximate a triangular transport map that bijectively maps between a tractable reference density and the high-fidelity target distribution. We introduce two strategies to leverage low-fidelity data: a hierarchical approach that composes maps between adjacent fidelity levels, and a non-hierarchical method that incorporates low-fidelity information through monotonicity-preserving corrections to the map parameterization. Numerical experiments compare these strategies with single-fidelity transport and demonstrate how the proposed multifidelity approaches can improve map estimation from limited high-fidelity data. To illustrate the broader utility of the learned maps, we also deploy them in a downstream amortized simulation-based inference task. This example shows that multifidelity improvements in map estimation can translate to improved conditional sampling and uncertainty quantification when high-fidelity data are scarce.

Given a polygon in the plane with vertices on lattice points, Pick's theorem expresses its area A in terms of the number I of lattice points in its interior and the number B of lattice points in its boundary: A = I + B/2 - 1. We consider polygons in the plane whose edges are halfway between lattice points. For such a polygon that is also unimodular, we express its area A in terms of the number I of lattice points in its interior and the number V of its vertices: A = I - V/8 + 1/2 .

Secure aggregation lets a server recover an aggregate of client updates without observing any individual update, but conventional protocols fix the aggregate precision when clients upload. We introduce and formulate a new progressive-resolution secure-aggregation functionality in which clients upload once and successively finer resolutions of the same aggregate can later be authorized without renewed client participation. To realize this functionality, we propose progressive-resolution secure aggregation (PSA): each clipped, dithered update is represented by compatible nested-lattice digits; separately releasable layers are protected by secure aggregation and an additional aggregate pad that remains unavailable to the server until a non-colluding release controller authorizes that layer.

We construct an extensive model of bulk mutual information (MI) in anti-de Sitter spacetime and use it to describe the leading quantum corrections to the MI of certain holographic theories. The equivalence between bulk and boundary MI follows from the Faulkner–Lewkowycz–Maldacena prescription. We show that the extensivity of the bulk MI is sufficient for the quantum-corrected boundary entropy vector to satisfy monogamy of MI and all other holographic entropy cone inequalities. Our model is built from a covariantly conserved current-current kernel in AdS and is realized explicitly by a free massive Dirac fermion, whose boundary MI at long distances is that of a fermionic generalized free field (GFF). The construction also reproduces the expected short-distance divergence structure of a GFF, and the current-current correlator defines a positive and monotonically decreasing function along the renormalization-group flow. We illustrate this explicitly for disjoint half-disks at the conformal fixed point of the model in AdS$_3$, where we present the exact result.

We study fixed-time input-to-state stability (FxT-ISS) for discrete-time systems subject to exogenous inputs using finite-horizon Lyapunov functions. In the proposed Lyapunov- based characterization, the effect of the initial condition is eliminated after a uniformly bounded number of iterations. This characterization encompasses existing extinction-based Lyapunov conditions while allowing more general finite-memory mechanisms. Building on this framework, we establish a finite-horizon small-gain theorem for interconnected discrete-time systems, providing conditions under which FxT-ISS is preserved under feedback interconnections. The proposed framework is illustrated through examples arising in distributed algorithms, optimization, and feedback optimization.

We study fixed-time input-to-state stability (FxT-ISS) for discrete-time systems subject to exogenous inputs. We introduce Lyapunov-based sufficient conditions for FxT- ISS based on a class of dissipation functions with a fixed-time extinction property, which encompasses commonly used two- power constructions while allowing substantially more general decay mechanisms. Building on this framework, we establish a small-gain theorem for interconnected FxT-ISS systems in the presence of disturbances. The results are illustrated through several examples, including perturbed optimization algorithms and a discrete-time feedback-optimization architecture.

In this paper, we construct explicit formulas for the heat kernel of the Hodge Laplacian restricted to an invariant subspace of $1$-forms on the real hyperbolic space $\mathbb{H}^{n}$. Using warped product decompositions, the problem is reduced to component operators on specific subspaces, yielding precise closed-form expressions for both odd and even dimensions. We further establish Millson-type recurrence relations that connect heat kernels across differing dimensions. Based on these explicit representations, we derive sharp uniform estimates and long-time asymptotic behavior, extending classical scalar heat kernel bounds to $1$-forms on real hyperbolic spaces.

We study ground states, criticality, and variational compactness for genuinely multipolar $L^p$-Hardy weights, $1<p<n$. A ground-state representation across the poles gives a complete criticality classification of the full potential and characterizes finite-energy attainment. Removing a geometric interaction term produces a variational problem governed by concentration at the poles and infinity. A strict concentration gap yields compactness, positive extremizers, and criticality of the optimal operator; for $p\ge2$, it is also necessary for attainment. In the subquadratic range, the optimal truncated coefficient is below one and attained throughout the full-potential critical set. For admissible parameters with $1<p<2$ and $(n-p)/p<s\le m(n-p)/p$, arbitrary fixed pole configurations exhibit negative directions, attained critical states, and coercive subcritical regimes outside this set along positive-weight paths towards the simplex vertices. A single-pole degeneration formula and continuity in the weights establish this coexistence. In the superquadratic range, an interaction threshold determines a non-attained infinity-concentration interval. Under strict separation from pole concentration, its endpoint is critical with infinite energy and a compact phase follows. We prove separation for every bipolar configuration and in several multipolar regimes, including a vertex neighborhood uniform in the exponent. Near parameters strictly inside the quadratic attained region, the optimal coefficient crosses one continuously through $p=2$. The proofs combine ground-state remainders, capacity, weighted compactness, and critical Hardy barriers.

We prove in ZFC that $\mathfrak d=\omega_1$ implies $\mathfrak a=\omega_1$, settling a question of Roitman from the 1970s. Here $\mathfrak d$ is the least size of a family that eventually dominates every function in $\omega^\omega$, and $\mathfrak a$ is the least size of an infinite maximal almost disjoint (MAD) family of infinite subsets of $\omega$. Consequently, $\mathfrak a\leq\mathfrak d$ whenever the continuum $\mathfrak c=2^{\aleph_0}$ is at most $\omega_2$. This answers negatively Shelah's question whether $\mathfrak d<\mathfrak a$ is consistent with $\mathfrak c=\omega_2$, and shows that his model of $\mathfrak d=\omega_2<\mathfrak a=\mathfrak c=\omega_3$ has the least possible values of $\mathfrak d$, $\mathfrak a$ and $\mathfrak c$ among models of $\mathfrak d<\mathfrak a$. The proof also gives a preservation theorem: from any dominating family of size $\omega_1$, we construct a MAD family of size $\omega_1$ that remains maximal in every outer model in which the given family is still dominating. Thus, under $\mathfrak d=\omega_1$, some MAD family is indestructible by every $\omega^\omega$-bounding forcing; it can moreover be chosen almost strongly separable, hence Cohen-indestructible. These results give negative answers, under $\mathfrak d=\omega_1$, to two questions of Hrušák and to one of Brendle, Guzmán, Hrušák and Raghavan.

We study non-autonomous basins of $F_j$ automorphisms of $\mathbb C^2$ satisfying $$a_j\|p\|\leq\|F_j(p)\|\leq b_j\|p\|\qquad (\forall p\in \mathbb B)$$ where $0<a_j<b_j<b<1$. We show that if $b_j^m<a_j$ for some integer $m$ and all $j>0$ then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have $\prod_{i=1}^{j-1}b_i\leq b_j^m$ then the basin is biholomorphic to $\mathbb C^2$. For the quadratic Hénon family $$F_n(z,w)=\bigl(z^2+c^{q_n}w,c^{q_n}z\bigr), \qquad 0<c<1,$$ we study how the asymptotic behavior of the sequence $q_n\to\infty$ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to $\mathbb C^2$ or is a Short $\mathbb C^2$, thereby isolating possible sequences for which other complex structures may occur.

We show that the Wasserstein space of the circle is not flat, in contrast with the flatness of the Wasserstein spaces of the real line and of an interval. Its sectional curvature is governed by a symplectic form on the Fourier coefficients, so that a plane is flat exactly when it is isotropic for that form, and on the plane spanned by the $n$-th Fourier modes the curvature is bounded above by $3n^2$, with equality exactly at the uniform measure. It is therefore unbounded. We compute it inside the formal Riemannian structure of Otto and confirm it directly from the definition of $W_2$, which also shows that no neighbourhood of the uniform measure in $P_2(S^1)$ is CAT(0). The result corrects the widely stated expectation that a Wasserstein space is flat whenever its base manifold is one-dimensional. The computation is carried out in the global frame of the tangent bundle of $P^\infty(M)$ induced by the eigenfunctions of the Laplacian of $M$, whose existence was observed by Lott. We prove that it is a Schauder basis of every tangent space and write Otto's metric and its Levi-Civita connection in it. On the circle this gives the metric and connection coefficients of $P^\infty(S^1)$ in closed form, as finite combinations of the Fourier coefficients of the density. A final section puts the frame to work on a computation from dynamics, reading the derivative of the push-forward action of an expanding circle map at its invariant measure as a weighted shift on the Fourier frequencies.

We examine how local motif structure and global network topology jointly shape spiking dynamics in stochastic neuronal networks. Using networks of Izhikevich neurons with Erdős-Rényi (ER) and scale-free (SF) background connectivity, we compare motif-embedded networks with synapse-count-matched, non-motif controls under noise- and stimulus-driven protocols. Motif embedding increases noise-induced coherence in both topologies and provides a smaller improvement in signal transmission, while SF-based networks attain greater absolute coherence and transmit signals faster and more reliably than ER-based networks. Motif-rewiring experiments show that this benefit depends on the specific arrangement of intra-motif connections rather than on strong local connectivity alone: randomizing the motif skeleton reduces coherence overall, although rewiring the non-recurrent feed-forward loop and bi-parallel motif can increase it. Targeted ablation shows that the SF advantage depends in part on hub integrity; removing high-degree excitatory neurons impairs coherence and signal transmission more than matched random node or edge removal. Together, these results indicate that local motifs and global hubs make distinct, complementary contributions to coherent spiking dynamics.

We study total path length in recursive trees with positive deterministic attachment weights. Writing $W_n=\sum_{i=1}^n w_i$ and $p_n=w_n/W_n$, we obtain exact martingale-innovation identities and a variance recurrence. Under the condition $p_n=O(n^{-1})$, centered total path length divided by $n$ converges almost surely and in $L^2$ to a nondegenerate random variable, and its variance is asymptotic to a positive constant times $n^2$. No polynomial asymptotic for $W_n$ is required. For power weights $w_i=i^\alpha$, the same argument applies to every real $\alpha$, including the critical and summable regimes beyond the positive-power cumulative-weight assumptions of existing profile theory. The expected average depth is logarithmic for $\alpha>-1$, iterated logarithmic for $\alpha=-1$, and bounded for $\alpha<-1$, while the global fluctuation scale remains linear throughout. In the summable regime we identify the random limit through the weighted depths of the infinite tree. The uniform case recovers the classical variance coefficient $2-\pi^2/6$. For linear weights we evaluate the coefficient as $8-2\pi^2/3$. Although this tree and a random binary search tree have identical insertion-depth marginals and expected total path length, their asymptotic variance coefficients differ by one. This gives an explicit comparison of global dependence that is invisible in individual depth distributions.

In this paper, we give exact combinatorial expressions for all CUE joint moments on the unit circle, in the case that all exponents are even positive integers, yielding the corresponding asymptotics. We offer two equivalent expressions for these quantities: the first is a sum involving Kostka numbers; the second is related to counting integer matrices (magic squares). As an application of our results, we unify many results in the literature, which were previously disconnected. Moreover, we prove a conjecture of Basor et al. (2018) regarding an irregular conformal block structure.

Neuronal networks exhibit complex dynamics shaped by connectivity and stochastic input. Empirical studies show that neuronal networks contain recurring subgraphs, or motifs, but the collective influence of different motif types after embedding in large stochastic networks remains less well understood. We construct a spiking network composed of six representative structural classes and examine how intrinsic noise, coupling strength, inter-motif connectivity, network size, and neuronal heterogeneity shape synchronization. Both motif- and network-level coherence peak at intermediate noise intensities, consistent with coherence resonance. Bidirectionally coupled pairs (M2) and the type-2 recurrent feed-forward loop (M3c) consistently exhibit high motif-level coherence. Rewiring these motifs while preserving local synapse number and strength produces some of the largest reductions in network coherence, showing that connection arrangement contributes beyond strong local coupling alone. M2 and M3c also exhibit frequent spike doublets, linking short inter-spike intervals with elevated coherence. Relative to a degree- and weight-matched random control, the motif-structured network reaches greater coherence at weaker noise. Increasing network size enhances coherence until the response begins to saturate, whereas applied-current heterogeneity lowers peak coherence and shifts the optimum toward stronger noise without erasing the relative differences among motif classes. These results show that local connection arrangement remains dynamically consequential after embedding and can shape noise-driven coherence at the population level.

This article is the third in a series investigating ill-posedness in critical Sobolev spaces for Camassa-Holm-type equations. Following results for the $b$-Novikov and Fokas-Olver-Rosenau-Qiao equations, we study the hyperelastic rod equation at the critical Sobolev regularity $H^{3/2}(\mathbb R)$. We prove ill-posedness at this endpoint, complementing the local well-posedness theory in $H^s(\mathbb R)$ for $s>3/2$ and the recently established norm-inflation result for $1<s<3/2$. The nature of the critical ill-posedness depends on the parameter $\gamma$. For $0<\gamma\leq3$, we construct arbitrarily small smooth initial data whose corresponding solutions develop finite-time blow-up, with the maximal lifespan tending to zero with the size of the initial data. For $\gamma<0$ or $\gamma>3$, we construct solutions with arbitrarily small $H^{3/2}$ initial data whose $H^{3/2}$ norm becomes arbitrarily large in arbitrarily short time.

Let $X$ be a nodal curve, and $G$ be the graph dual to $X$. A stability condition on a graph $G$ is an assignment of integers to the biconnected subsets of vertices of $G$ satisfying some desired properties. Stability conditions yield degeneracy sets, which are certain collections of biconnected subsets of $G$, but degeneracy sets can also be described without a reference stability condition. We show that the degeneracy sets of a graph $G$ correspond to the single-element extensions of the graphic matroid $M(G)$. Some single-element extensions of the graphic matroid $M(G)$ can be oriented to be single-element extensions of the oriented graphic matroid $\mathcal{M}(G)$. Single-element extensions of oriented matroids are in bijection with subdivisions of the Lawrence polytope of the dual oriented matroid. We construct stability conditions on $G$ from subdivisions of the Lawrence polytope of the cographic matroid $\mathcal{M}^\ast(G)$. Finally, we show that any degeneracy set $\mathcal{D}$ of a graph $G$ that has the set of all biconnected subsets of $G$ as a lower bound in the poset of degeneracy sets corresponds to an orientable extension of $M(G)$.

In this paper, in addition to establishing several new sufficient or equivalent conditions for the well-known Separable Quotient Problem (SQP), we show that every infinite-dimensional Banach space lacking the Lotz property admits a separable quotient, thereby revealing a surprising connection between the SQP and semigroup theory.

Motivated by John Baez's question about seven exceptional coincidences among binomial coefficients, we construct seven exterior-power identities for representations of $A_5$. Their common symmetry comes from simplex edges and icosahedral vertices, axes, and faces, together with characters of stabilizers. We enumerate all ordered pairs of representation isomorphism classes in the prescribed dimensions. Apart from the pair of trivial actions, the permutation pair at $1540$ and the complex monomial pair at $7140$ are unique. At $3003$, Klein's twelve-point divisor gives a noncanonical intertwiner factored through polynomial division, differentiation, and residues.

Large reasoning models (LRMs) have achieved substantial improvements in solving complex mathematical problems, but often produce lengthy, repetitive, or erroneous reasoning trajectories. In this work, we introduce a new interpretability framework, SLIDER, to evaluate the quality of the reasoning process. SLIDER leverages an emerging body of work from information theory called Partial Information Decomposition to disentangle the information about the final answer between two consecutive reasoning steps into non-negative components: unique information (in preceding steps or current step), redundant information, and synergistic information. Building on this decomposition, we propose the *Step-wise Repetitive Reasoning Index (Step-RRI)*, a theoretically grounded measure that assesses whether the answer-relevant information in the current step $S_i$ is predominantly redundant with the past steps $S_{<i}$, relative to its unique and synergistic contributions. To evaluate the effectiveness of Step-RRI in detecting repetitiveness, we apply SLIDER to the redundancy class of the PRMBench dataset where Step-RRI improves step-level redundancy identification accuracy by over $10$ points compared to embedding-similarity and information-gain baselines. Next, we define *Trajectory-RRI*, an aggregate measure of repetitiveness for an individual reasoning trajectory. To demonstrate its practical relevance, we show that average Trajectory-RRI strongly correlates with actual reasoning length across QwQ-32B, DeepSeek-R1-Distill-Qwen-32B, and GPT-4.1, motivating its use as a signal for improving reasoning efficiency. Finally, we introduce *Trajectory-RRI-guided data selection for fine-tuning*, demonstrating that selecting training data based on Trajectory-RRI can improve a fine-tuned model's reasoning efficiency while largely preserving its task performance.

A domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a round circle or a single point. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. Conformal rigidity is closely related to Koebe's conjecture, which asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. In this article, we survey recent progress on conformal rigidity, with particular emphasis on its relationship with conformal removability.

We introduce a conservative two-exponent dispersive family, with $p>1$ the nonlinear-flux exponent and $r>0$ the gradient-dispersion exponent, that separates the mechanisms governing compactification, free-boundary regularity and linearised spectral structure while retaining an explicit travelling-wave theory. The family reduces to the absolute Rosenau-Hyman equation when $r=1$. The deformation leaves the amplitude-speed law unchanged, but makes the width amplitude dependent except at $r=1$. Two independent transition curves emerge: $pr=1$ separates infinite-tail solitary waves from compactons, whereas $r(p-1)=2$ separates compactons with a regular free boundary from edge-cusped compactons with divergent slope. The latter remain distributional solutions, with no singular measures at either the free boundary or the crest. For $r\ge1$, the linearisation factors through a singular Sturm-Liouville operator whose natural Friedrichs realisation is solvable: after a compacton-adapted change of variable, the eigenvalue equation is of Jacobi-hypergeometric type, with two explicit spectral sequences, exactly one negative eigenvalue and a one-dimensional translation kernel. Numerical experiments connect these exact structures to approximation and dynamics. Measured Fourier tails and projection errors agree with the rates predicted by the free-boundary exponent, while padded Fourier-Galerkin time integration propagates the cusped waves coherently with small phase drift. A structure-adapted Jacobi-Galerkin diagnostic of the full generator finds no detectable real part in the tested truncated point spectra. The family therefore provides an explicit testbed in which geometric singularity, weak admissibility, spectral structure and numerical regularity of compactons can be studied within a single degenerate dispersive model.